Finite Group Databasebeta

← C67×#20151 #30699402

Subgroups — C67×#20151 #30699402

Order 17152.

580 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3400C211yesmaximal, characteristic, complemented
2C67×#3396C211yesmaximal, characteristic
3C67×#3374C211yesmaximal, characteristic, complemented
4C67×#3365C211yesmaximal, characteristic, complemented
5C67×#1717C211yesmaximal, characteristic
6C67×#1702C211yesmaximal, characteristic, complemented
7C67×#1688C211yesmaximal, characteristic
8C67×#1680C211yesmaximal, characteristic, complemented
9C67×#1666C211yesmaximal, characteristic, complemented
10C67×#1640C211yesmaximal, characteristic, complemented
11C67×#1640C211yesmaximal, characteristic, complemented
12C67×#1626C211yesmaximal, characteristic, complemented
13C67×#1626C211yesmaximal, characteristic, complemented
14C67×#1587C211yesmaximal, characteristic, complemented
15C67×#1587C211yesmaximal, characteristic, complemented
16#559×C67C2211yescharacteristic
17#559×C67C2211yescharacteristic
18#557×C67C2211yescharacteristic
19#557×C67C2211yescharacteristic
20#497×C67—21no—
21#495×C67—21no—
22#493×C67C2211yescharacteristic
23#492×C67C2211yescharacteristic
24#491×C67—21no—
25#488×C67—21no—
26#487×C67C2211yescharacteristic
27#486×C67C2211yescharacteristic
28#486×C67C2211yescharacteristic
29#486×C67C2211yescharacteristic
30#479×C67—21no—
31#478×C67—21no—
32#475×C67C2211yescharacteristic
33#475×C67C2211yescharacteristic
34#464×C67—21no—
35#464×C67—21no—
36#463×C67—21no—
37#463×C67—21no—
38#462×C67—21no—
39#462×C67C2211yescharacteristic
40#462×C67C2211yescharacteristic
41#461×C67—21no—
42#460×C67C2211yescharacteristic
43#460×C67C2211yescharacteristic
44#460×C67C2211yescharacteristic
45#460×C67C2211yescharacteristic
46#446×C67C2211yescharacteristic
47#446×C67C2211yescharacteristic
48C8×Q8×C67C2211yescharacteristic
49C8×Q8×C67C2211yescharacteristic
50#441×C67—21no—
51#440×C67—21no—
52C4×Q16×C67C2211yescharacteristic
53C4×Q16×C67C2211yescharacteristic
54C4×SD16×C67—21no—
55C4×D8×C67—21no—
56#334×C67C2211yescharacteristic
57#334×C67C2211yescharacteristic
58#332×C67C2211yescharacteristic
59#332×C67C2211yescharacteristic
60#331×C67C2211yescharacteristic
61#331×C67C2211yescharacteristic
62Q8⋊C8×C67C2211yescharacteristic
63Q8⋊C8×C67C2211yescharacteristic
64Q8⋊C8×C67C2211yescharacteristic
65Q8⋊C8×C67C2211yescharacteristic
66#322×C67C2211yescharacteristic
67C2×Q16×C67—21no—
68C2×Q16×C67—21no—
69C2×Q16×C67D411yescharacteristic
70C2×Q16×C67D411yescharacteristic
71C2×SD16×C67—21no—
72C2×SD16×C67—21no—
73C2×SD16×C67—21no—
74C2×SD16×C67—21no—
75C2×SD16×C67—21no—
76C2×SD16×C67—21no—
77C2×D8×C67—21no—
78C2×D8×C67—21no—
79C4⋊Q8×C67—21no—
80C4⋊Q8×C67—21no—
81C4⋊Q8×C67C2311yescharacteristic
82C4⋊Q8×C67C2311yescharacteristic
83#127×C67C2311yescharacteristic
84#127×C67C2311yescharacteristic
85#125×C67—21no—
86#125×C67—21no—
87#125×C67—41no—
88#125×C67—41no—
89#124×C67—21no—
90#124×C67—21no—
91#124×C67—21no—
92#124×C67—21no—
93#123×C67—41no—
94#123×C67—41no—
95C4×Q8×C67—21no—
96C4×Q8×C67—21no—
97C4×Q8×C67—21no—
98C4×Q8×C67—21no—
99C4×Q8×C67C2311yescharacteristic
100C4×Q8×C67C2311yescharacteristic
101C4×Q8×C67C2311yescharacteristic
102C4×Q8×C67C2311yescharacteristic
103C4×D4×C67—21no—
104C4×D4×C67—21no—
105C4×D4×C67—21no—
106C4×D4×C67—21no—
107#107×C67D411yescharacteristic
108#107×C67D411yescharacteristic
109#106×C67—21no—
110#106×C67—21no—
111C4⋊C8×C67—21no—
112C4⋊C8×C67—21no—
113C4⋊C8×C67—21no—
114C4⋊C8×C67—21no—
115C4⋊C8×C67C2311yescharacteristic
116C4⋊C8×C67C2311yescharacteristic
117C4⋊C8×C67C2311yescharacteristic
118C4⋊C8×C67C2311yescharacteristic
119#103×C67—21no—
120#103×C67—21no—
121#103×C67—21no—
122#103×C67—21no—
123#103×C67—21no—
124#103×C67—21no—
125#103×C67D411yescharacteristic
126#103×C67D411yescharacteristic
127#103×C67D411yescharacteristic
128#103×C67D411yescharacteristic
129#102×C67—21no—
130#102×C67—21no—
131#102×C67—21no—
132#102×C67—21no—
133#102×C67—21no—
134#102×C67—21no—
135#102×C67—41no—
136#102×C67—41no—
137#97×C67—21no—
138#97×C67—21no—
139C4×C8×C67—21no—
140C4×C8×C67—21no—
141C4×C8×C67C2311yescharacteristic
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C2×Q8×C67—21no—
145C2×Q8×C67—21no—
146C2×Q8×C67C2×D411yescharacteristic
147C2×Q8×C67C2×D411yescharacteristic
148C2×Q8×C67C2×D411yescharacteristic
149C2×Q8×C67C2×D411yescharacteristic
150C2×D4×C67—21no—
151C2×D4×C67—21no—
152C2×D4×C67—21no—
153C2×D4×C67—21no—
154C2×D4×C67—21no—
155C2×D4×C67—21no—
156C22×C4×C67—41no—
157C22×C4×C67—41no—
158Q16×C67—41no—
159Q16×C67—41no—
160Q16×C67—42no—
161Q16×C67—81no—
162SD16×C67—42no—
163SD16×C67—42no—
164SD16×C67—81no—
165SD16×C67—81no—
166SD16×C67—81no—
167SD16×C67—81no—
168D8×C67—42no—
169D8×C67—81no—
170C2×C8×C67—21no—
171C2×C8×C67—21no—
172C2×C8×C67—21no—
173C2×C8×C67—21no—
174C2×C8×C67—21no—
175C2×C8×C67C2×D411yescharacteristic
176C2×C8×C67C2×D411yescharacteristic
177C2×C8×C67C2×D411yescharacteristic
178C2×C8×C67C2×D411yescharacteristic
179C4⋊C4×C67—21no—
180C4⋊C4×C67—21no—
181C4⋊C4×C67—21no—
182C4⋊C4×C67—21no—
183C4⋊C4×C67—21no—
184C4⋊C4×C67—21no—
185C4⋊C4×C67—21no—
186C4⋊C4×C67—21no—
187C4⋊C4×C67—41no—
188C4⋊C4×C67—41no—
189C4⋊C4×C67—41no—
190C4⋊C4×C67—41no—
191C4⋊C4×C67—41no—
192C4⋊C4×C67—41no—
193C4⋊C4×C67C2×D411yescharacteristic
194C4⋊C4×C67C2×D411yescharacteristic
195C4⋊C4×C67C2×D411yescharacteristic
196C4⋊C4×C67C2×D411yescharacteristic
197C22⋊C4×C67—41no—
198C22⋊C4×C67—41no—
199C22⋊C4×C67—41no—
200C22⋊C4×C67—41no—
201C22⋊C4×C67—41no—
202C22⋊C4×C67—41no—
203C42×C67—21no—
204C42×C67—21no—
205C42×C67—21no—
206C42×C67—21no—
207C42×C67—41no—
208C42×C67—41no—
209C42×C67C2411yescharacteristic
210C23×C67—41no—
211C23×C67—41no—
212Q8×C67—21no—
213Q8×C67—21no—
214Q8×C67—21no—
215Q8×C67—21no—
216Q8×C67—21no—
217Q8×C67—21no—
218Q8×C67—21no—
219Q8×C67—21no—
220Q8×C67—41no—
221Q8×C67—41no—
222Q8×C67—41no—
223Q8×C67—41no—
224D4×C67—41no—
225D4×C67—41no—
226D4×C67—41no—
227D4×C67—41no—
228D4×C67—41no—
229D4×C67—41no—
230D4×C67—41no—
231D4×C67—41no—
232D4×C67—81no—
233D4×C67—81no—
234C2×C4×C67—21no—
235C2×C4×C67—21no—
236C2×C4×C67—21no—
237C2×C4×C67—21no—
238C2×C4×C67—21no—
239C2×C4×C67—21no—
240C2×C4×C67—21no—
241C2×C4×C67—21no—
242C2×C4×C67—41no—
243C2×C4×C67—41no—
244C2×C4×C67—161no—
245C2×C4×C67—161no—
246C2×C4×C67C22×D411yescharacteristic
247C2×C4×C67C22×D411yescharacteristic
248C2×C4×C67H(5,𝔽2)11yescharacteristic
249C8×C67—21no—
250C8×C67—21no—
251C8×C67—21no—
252C8×C67—21no—
253C8×C67—41no—
254C8×C67—41no—
255C8×C67—41no—
256C8×C67—41no—
257C8×C67—41no—
258C22×C67—81no—
259C22×C67—81no—
260C22×C67—81no—
261C22×C67—81no—
262C22×C67—81no—
263C22×C67—81no—
264C22×C67D4211yescharacteristic
265C4×C67—21no—
266C4×C67—41no—
267C4×C67—41no—
268C4×C67—41no—
269C4×C67—41no—
270C4×C67—41no—
271C4×C67—41no—
272C4×C67—41no—
273C4×C67—41no—
274C4×C67—81no—
275C4×C67—81no—
276C4×C67#57511yescharacteristic
277C4×C67#57511yescharacteristic
278C4×C67#57711yescharacteristic
279C4×C67#57711yescharacteristic
280#20151C6711yesmaximal, characteristic, direct product factor, Sylow
281C2×C67—161no—
282C2×C67—161no—
283C2×C67#327111yescharacteristic
284C2×C67#327111yescharacteristic
285C2×C67#329011yescharacteristic
286#3400C2×C6711yescharacteristic, complemented
287#3396C2×C6711yescharacteristic
288#3374C2×C6711yescharacteristic, complemented
289#3365C2×C6711yescharacteristic, complemented
290#1717C2×C6711yescharacteristic
291#1702C2×C6711yescharacteristic, complemented
292#1688C2×C6711yescharacteristic
293#1680C2×C6711yescharacteristic, complemented
294#1666C2×C6711yescharacteristic, complemented
295#1640C2×C6711yescharacteristic, complemented
296#1640C2×C6711yescharacteristic, complemented
297#1626C2×C6711yescharacteristic, complemented
298#1626C2×C6711yescharacteristic, complemented
299#1587C2×C6711yescharacteristic, complemented
300#1587C2×C6711yescharacteristic, complemented
301C67#2015111yescharacteristic, direct product factor, Sylow
302#559C22×C6711yescharacteristic
303#559C22×C6711yescharacteristic
304#557C22×C6711yescharacteristic
305#557C22×C6711yescharacteristic
306#497—21no—
307#495—21no—
308#493C22×C6711yescharacteristic
309#492C22×C6711yescharacteristic
310#491—21no—
311#488—21no—
312#487C22×C6711yescharacteristic
313#486C22×C6711yescharacteristic
314#486C22×C6711yescharacteristic
315#486C22×C6711yescharacteristic
316#479—21no—
317#478—21no—
318#475C22×C6711yescharacteristic
319#475C22×C6711yescharacteristic
320#464—21no—
321#464—21no—
322#463—21no—
323#463—21no—
324#462—21no—
325#462C22×C6711yescharacteristic
326#462C22×C6711yescharacteristic
327#461—21no—
328#460C22×C6711yescharacteristic
329#460C22×C6711yescharacteristic
330#460C22×C6711yescharacteristic
331#460C22×C6711yescharacteristic
332#446C22×C6711yescharacteristic
333#446C22×C6711yescharacteristic
334C8×Q8C22×C6711yescharacteristic
335C8×Q8C22×C6711yescharacteristic
336#441—21no—
337#440—21no—
338C4×Q16C22×C6711yescharacteristic
339C4×Q16C22×C6711yescharacteristic
340C4×SD16—21no—
341C4×D8—21no—
342#334C22×C6711yescharacteristic
343#334C22×C6711yescharacteristic
344#332C22×C6711yescharacteristic
345#332C22×C6711yescharacteristic
346#331C22×C6711yescharacteristic
347#331C22×C6711yescharacteristic
348Q8⋊C8C22×C6711yescharacteristic
349Q8⋊C8C22×C6711yescharacteristic
350Q8⋊C8C22×C6711yescharacteristic
351Q8⋊C8C22×C6711yescharacteristic
352#322C22×C6711yescharacteristic
353C2×Q16—21no—
354C2×Q16—21no—
355C2×Q16D4×C6711yescharacteristic
356C2×Q16D4×C6711yescharacteristic
357C2×SD16—21no—
358C2×SD16—21no—
359C2×SD16—21no—
360C2×SD16—21no—
361C2×SD16—21no—
362C2×SD16—21no—
363C2×D8—21no—
364C2×D8—21no—
365C4⋊Q8—21no—
366C4⋊Q8—21no—
367C4⋊Q8C23×C6711yescharacteristic
368C4⋊Q8C23×C6711yescharacteristic
369#127C23×C6711yescharacteristic
370#127C23×C6711yescharacteristic
371#125—21no—
372#125—21no—
373#125—41no—
374#125—41no—
375#124—21no—
376#124—21no—
377#124—21no—
378#124—21no—
379#123—41no—
380#123—41no—
381C4×Q8—21no—
382C4×Q8—21no—
383C4×Q8—21no—
384C4×Q8—21no—
385C4×Q8C23×C6711yescharacteristic
386C4×Q8C23×C6711yescharacteristic
387C4×Q8C23×C6711yescharacteristic
388C4×Q8C23×C6711yescharacteristic
389C4×D4—21no—
390C4×D4—21no—
391C4×D4—21no—
392C4×D4—21no—
393#107D4×C6711yescharacteristic
394#107D4×C6711yescharacteristic
395#106—21no—
396#106—21no—
397C4⋊C8—21no—
398C4⋊C8—21no—
399C4⋊C8—21no—
400C4⋊C8—21no—
401C4⋊C8C23×C6711yescharacteristic
402C4⋊C8C23×C6711yescharacteristic
403C4⋊C8C23×C6711yescharacteristic
404C4⋊C8C23×C6711yescharacteristic
405#103—21no—
406#103—21no—
407#103—21no—
408#103—21no—
409#103—21no—
410#103—21no—
411#103D4×C6711yescharacteristic
412#103D4×C6711yescharacteristic
413#103D4×C6711yescharacteristic
414#103D4×C6711yescharacteristic
415#102—21no—
416#102—21no—
417#102—21no—
418#102—21no—
419#102—21no—
420#102—21no—
421#102—41no—
422#102—41no—
423#97—21no—
424#97—21no—
425C4×C8—21no—
426C4×C8—21no—
427C4×C8C23×C6711yescharacteristic
428C4×C8C23×C6711yescharacteristic
429C4×C8C23×C6711yescharacteristic
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8C2×D4×C6711yescharacteristic
433C2×Q8C2×D4×C6711yescharacteristic
434C2×Q8C2×D4×C6711yescharacteristic
435C2×Q8C2×D4×C6711yescharacteristic
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C22×C4—41no—
443C22×C4—41no—
444Q16—41no—
445Q16—41no—
446Q16—42no—
447Q16—81no—
448SD16—42no—
449SD16—42no—
450SD16—81no—
451SD16—81no—
452SD16—81no—
453SD16—81no—
454D8—42no—
455D8—81no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8C2×D4×C6711yescharacteristic
462C2×C8C2×D4×C6711yescharacteristic
463C2×C8C2×D4×C6711yescharacteristic
464C2×C8C2×D4×C6711yescharacteristic
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4C2×D4×C6711yescharacteristic
480C4⋊C4C2×D4×C6711yescharacteristic
481C4⋊C4C2×D4×C6711yescharacteristic
482C4⋊C4C2×D4×C6711yescharacteristic
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—41no—
494C42—41no—
495C42C24×C6711yescharacteristic
496C23—41no—
497C23—41no—
498Q8—21no—
499Q8—21no—
500Q8—21no—
501Q8—21no—
502Q8—21no—
503Q8—21no—
504Q8—21no—
505Q8—21no—
506Q8—41no—
507Q8—41no—
508Q8—41no—
509Q8—41no—
510D4—41no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—41no—
517D4—41no—
518D4—81no—
519D4—81no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—161no—
531C2×C4—161no—
532C2×C4C22×D4×C6711yescharacteristic
533C2×C4C22×D4×C6711yescharacteristic
534C2×C4H(5,𝔽2)×C6711yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—41no—
540C8—41no—
541C8—41no—
542C8—41no—
543C8—41no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22D42×C6711yescharacteristic
551C4—21no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—81no—
561C4—81no—
562C4#575×C6711yescharacteristic
563C4#575×C6711yescharacteristic
564C4#577×C6711yescharacteristic
565C4#577×C6711yescharacteristic
566C2—161no—
567C2—161no—
568C2C67×#327111yescharacteristic
569C2C67×#327111yescharacteristic
570C2C67×#329011yescharacteristic
subgroupG/H
center, socleC22×C67D42
commutator, Frattini, cosocleC42C24×C67
prime radical#493×C67C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C67 ◁ C42×C67 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C67×#3400C2semidirect
3. C67×#3374C2semidirect
4. C67×#3365C2semidirect
6. C67×#1702C2semidirect
8. C67×#1680C2semidirect
9. C67×#1666C2semidirect
10. C67×#1640C2semidirect
11. C67×#1640C2semidirect
12. C67×#1626C2semidirect
13. C67×#1626C2semidirect
14. C67×#1587C2semidirect
15. C67×#1587C2semidirect
280. #20151C67direct (N × H)
286. #3400C2×C67semidirect
288. #3374C2×C67semidirect
289. #3365C2×C67semidirect
291. #1702C2×C67semidirect
293. #1680C2×C67semidirect
294. #1666C2×C67semidirect
295. #1640C2×C67semidirect
296. #1640C2×C67semidirect
297. #1626C2×C67semidirect
298. #1626C2×C67semidirect
299. #1587C2×C67semidirect
300. #1587C2×C67semidirect
301. C67#20151direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C67×#34002
C67×#33962
C67×#33742
C67×#33652
C67×#17172
C67×#17022
C67×#16882
C67×#16802
C67×#16662
C67×#16402
C67×#16262
C67×#15872
#2015167